A non-commutative unitary analogue of Kirchberg’s conjecture

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Abstract

The C-algebra Unc(n) is the universal C-algebra generated by n2 generators uij that make up a unitary matrix. We prove that Kirchberg’s formulation of Connes’ embedding problem has a positive answer if and only if Unc(2)min Unc(2) = Unc(2)max Unc(2). Our results follow from properties of the finite-dimensional operator system Vn spanned by 1 and the generators of Unc(n). We show that Vn is an operator system quotient of M2n and has the OSLLP. We obtain necessary and sufficient conditions on Vn for there to be a positive answer to Kirchberg’s problem. Finally, in analogy with recent results of Ozawa, we show that a form of Tsirelson’s problem related to Vn is equivalent to Connes’ embedding problem.

Original languageEnglish (US)
Pages (from-to)503-536
Number of pages34
JournalIndiana University Mathematics Journal
Volume68
Issue number2
DOIs
StatePublished - 2019
Externally publishedYes

Keywords

  • Connes’ embedding problem
  • Kirchberg’s conjecture
  • Operator systems
  • Unitary correlation sets

ASJC Scopus subject areas

  • General Mathematics

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